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About headwrong

Ah, sorry - I didn't realise it was an installer. I thought it was just the standalone app that could be run side-by-side with the mainstream iTunes. It won't overwrite your library but it might be a pain in the arse to get it back to the proper version of iTunes.
Not sure how to help here, sorry.

In a bit of a convoluted way, yes
You'll have to download this version of iTunes that still has the App Store built in: https://secure-appldnld.apple.com/itunes12/091-60759-201803029-1F70CB08-3131-11E8-9791-31052B2AA206/iTunes12.6.4.dmg
You'll then be able to purchase Designer and it will be safely waiting in your Purchased app list on your new iPad, when you get it.